Representation theory and the X-ray transform

Transcription

1 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 1/18 Representation theory and the X-ray transform Differential geometry on real and complex projective space Michael Eastwood Australian National University

2 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 2/18 Topics Connections Affine connections Levi-Civita connection Round sphere and real projective space Complex projective space Fubini-Study curvature Model embeddings Kähler form and model pullback Representation theory Tensors under model pullback Pullback of curvature et cetera

3 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 3/18 Connections E= smooth vector bundle (real or complex) Connection : E Λ 1 E Coupled de Rham E s.t. ( fσ)= f σ+d f σ Λ 1 E Λ 2 } {{ E} Λ n E 0 ω σ dω σ ω σ Curvature 2 : E Λ 2 E 2 ( fσ)= f 2 σ κ Γ(Λ 2 End(E)) Bianchi identity: 3 = 0 or κ=0

4 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 4/18 Induced connections If E and F are equipped with connections, then there are induced connections on the following bundles. E E E E E E E End(E)= E E E F Hom(E, F)= E F and so on... Leibniz rule, e.g. (σ τ)=( σ) τ+σ ( τ)

5 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 5/18 Existence and freedom, ˆ connections on E h + (1 h) ˆ also a connection. Therefore, partition of unity existence. Freedom:, ˆ connections on E ˆ = +Γ forγ : E Λ 1 E a homomorphism.

6 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 6/18 Affine connections Affine connection onλ 1 (or the tangent bundle) Problem: :Λ 1 Λ 1 Λ 1 Λ 2 may not agree with the exterior derivative! Solution: T d :Λ 1 Λ 2 Λ 1 Λ 1 is a homomorphism torsion Γ(Λ 1 End(Λ 1 )). ˆ T is a torsion free connection onλ 1. Consequence:Λ 1 Λ 1 ˆ Λ 2 Λ 1 is unambiguous.

7 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 7/18 Indices Covariant tensorsφ a,ψ ab,.... (Anti)-symmetrisation: φ [ab] 1 2 (φ ab φ ba ) φ (ab)c 1 2 (φ abc+φ bac ) Contravariant tensors, e.g. X a = vector field. E.g. torsion tensor T ab c s.t. T ab c = T [ab] c or T (ab) c = 0 Einstein summation convention: X ω X a ω a Curvature of a torsion-free affine connection ( a b b a )X c c = R ab d X d c ( a b b a )ω d = R ab d ω c

8 Levi-Civita connection Given g ab a metric,! affine a characterised by a is torsion-free a g bc = 0. Proof Choose any ˆ a torsion-free. Consider a φ b = ˆ a φ b Γ ab c φ c. Want: Γ abc =Γ (ab)c andγ a(bc) = 1 2 ˆ a g bc but Λ 1 Λ 2 Λ 2 Λ 1 NB K abc K [ab]c QED! K a[bc] AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 8/18

9 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 9/18 Round sphere Riemannian curvature tensor ( a b b a )ω d = R ab c d ω c R abcd = R [ab][cd] R [abc]d = 0 ( R abcd = R cdab ) = SO(n+1)/SO(n) R abcd = g ac g bd g bc g ad R ab R ca c b = (n 1)g ab constant curvature R R a a = n(n 1)

10 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 10/18 Real projective space antipodal identification = SO(n+1)/S(O(1) O(n)) R abcd = g ac g bd g bc g ad constant curvature i.e. ( b c c b )ω a = g ab ω c g ac ω b RP n ={linear L R n+1 s.t. dim L=1}

11 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 11/18 Complex projective space CP n RP n = {linear L C n+1 s.t. dim L=1} = SU(n+1)/S(U(1) U(n)) Fubini-Study metric g ab complex span CP n totally geodesic embedding J b a s.t. J b a J c c b = δ a complex structure (orthogonal) J ab ( J c a g bc ) Kähler form (skew) a J bc = 0 Fubini-Study curvature R abcd = g ac g bd g bc g ad + J ac J bd J bc J ad + 2J ab J cd NB: R ab c d J ce = 2J a(d g e)b 2J b(d g e)a + 2J ab g de

12 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 12/18 Model embeddings ι Recall totally geodesic embeddingrp n CP n Recall SU(n+1) acting oncp n by isometries Model embeddings:rp n µ CP n. J is type (1, 1) ι J= 0. Thereforeµ J= 0, i.e. model embeddings are Lagrangian. Conversely, Model embeddings though p CP n Linear Algebra Lagrangian subspaces of T p CP n ( Lagrangian Grassmannian U(n)/O(n) E.g. Helgason Geometric Analysis..., Exercise I.A.4(ii) )

13 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 13/18 Kähler form a J bc = 0 dj=0 J ab is non-degenerate } CP n is a symplectic manifold Letψbe a 2-form, i.e. with indicesψ ab =ψ [ab]. Then ψ ab = ψ ab 1 2n Jcd ψ cd J ab + 1 2n Jcd ψ cd J ab } {{ } J-trace-free ψ ab = ψ ab +θj ab ψ = ψ +θj Λ 2 = Λ 2 Λ 0 J Symplectic decomposition

14 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 14/18 Representation theory Λ 2 =Λ 2 Λ0 J (on any symplectic manifold) Let J be a non-degenerate skew 2n 2n real matrix. Sp(2n,R) {A=2n 2n matrix s.t. AJA t = J} Defining representation :R 2n Λ 2 R 2n =Λ 2 R 2n R = n 1 nodes n nodes n nodes Branching for SL(2n, R) Sp(2n, R). (Induced vector bundles from co-frame bundle.)

15 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 15/18 Model pullback Recall model embeddingsµ :RP n CP n. Supposeψis a two-form oncp n. Lemma: µ ψ=0 µ ψ=θj ψ = 0. Proof: Fix attention on p CP n. ψ(p) Λ 2 p s.t.ψ(p) L = 0 Lagrangian L T p CP n NB: invariant under Sp(2n,R) acting on T p CP n. Recallµ J= 0. ψ s.t.µ ψ 0. Schur! Generalises to suitable tensors, using a b c d = a b c d (case n=4).

16 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 16/18 Curvature pullback ψ abcd =ψ [ab][cd] ψ [abc]d = 0 } Riemann tensor symmetries ψ abcd Γ( ) (case n=4). Branch to SL(8, R) Sp(8, R) = ψ abcd = ψ abcd + φ ab J cd + θj ab J cd cf. Weyl cf. Ricci cf. Scalar Lemma:µ ψ abcd = 0 modelsµ ψ abcd = 0

17 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 17/18 Summary Curvature onrp n R abcd = g ac g bd g bc g ad Curvature oncp n R abcd = g ac g bd g bc g ad + J ac J bd J bc J ad + 2J ab J cd Model embeddings µ :RP n CP n totally geodesic 2-form lemma µ ψ ab = 0 µ ψ ab = 0 Curvature lemma µ ψ abcd = 0 µ ψ abcd = 0

18 AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 18/18 THANK YOU END OF PART ONE